The occurrence of real eigenvalue spectra in non-Hermitian
systems that are symmetric under combined
parity reflection \(\mathcal {P}\)  and time reversal \(\mathcal {T}\)  has, since its first
demonstration by Bender and Boettcher in 1998 [1]}, inspired a wide
variety of theoretical and experimental studies in \(\mathcal {PT}\) -symmetric physics.
While initially focusing on classical and semi-classical systems these
studies were soon extended to include bosonic field theories as well
[2]}.
For fermionic systems, Jones-Smith et al. [3]}
brought attention to the property of odd time-reversal symmetry
(\(\mathcal {T}^2=-\mathbb {1}\) ) in \(3+1\)  dimensions and its importance in
the context of \(\mathcal {PT}\)  symmetry,
in contrast to the even time-reversal symmetry of bosonic systems.
